On Optimal Encodings and Systematicity in Function-Correcting Codes

2026-10-06

AI Summary

This study addresses the high encoding redundancy of systematic functional error-correcting codes and the challenge of constructing optimal non-systematic encodings. By transcending conventional systematic constraints, this work investigates optimal encoding schemes for both linear and nonlinear functional error-correcting codes. Methodologically, it integrates information theory, combinatorial optimization, and simplicial order minimality theory to analyze code constructions. It demonstrates that non-systematic encoding substantially reduces redundancy and proposes an algorithm for constructing optimal generator matrices based on greedy bases. The primary contributions include reducing the encoding redundancy to one, establishing the existence conditions for two-valued functions along with their redundancy thresholds, and significantly enhancing the overall encoding efficiency of functional error-correcting codes.

Abstract

Function-correcting codes (FCCs) protect the value of a function $f$ of the message against $t$ errors. In the original formulation of Lenz et al. (2023), the encoding is systematic, and the lower bound of $2t$ on the redundancy relies on this form. Recently, D. Ho (arXiv, 2026) showed, for linear functions and linear encodings, that systematicity can cost redundancy. We begin with the OR function to show that the cost is not confined to the linear setting: a non-systematic encoding attains redundancy $1$ while every systematic encoding needs $2t$. For a linear function $f$ and a fixed linear code $C$, let $d_f$ denote the minimum distance between codewords of messages with different function values. Different generator matrices of $C$ assign different codewords to the messages and can give different values of $d_f$. We study which generator matrix of $C$ gives the largest $d_f$. We give an algorithm that constructs an optimal generator matrix and determines the optimal value as the weight of a codeword in a greedy basis of $C$. We then characterize, in terms of information sets, when a generator matrix in systematic form attains this optimum, and give a necessary condition that is checked on the low-weight codewords of $C$ alone. For two-valued functions, we drop both linearity and systematicity. Using the minimality of initial segments of the simplicial order with respect to Hamming neighbourhoods, we show that a non-systematic $(f,t)$-FCC of length $n$ exists if and only if a condition depending on $f$ only through the size of its smaller preimage holds. For $t=1$, redundancy $1$ is sufficient for every nonconstant two-valued function on $\mathbb{F}_2^k$ with $k\ge 10$. For general $t$, redundancy $1$ suffices for all two-valued functions once $k$ is large enough, and for each $s<2t$ we give an upper bound on the threshold in $k$ beyond which redundancy $s$ suffices.

Problem

• Function-correcting codes
• Systematicity
• Redundancy
• Optimal encodings
• Generator matrix

Innovation

• Function-correcting codes
• Non-systematic encoding
• Optimal generator matrix
• Simplicial order
• Redundancy optimization